The integral of the characteristic function is zero, but the probability of finding a rational in any interval is one. In Feynman’s argument, it seems he was re

The integral of the characteristic function is zero, but the probability of finding a rational in any interval is one. In Feynman’s argument, it seems he was relying on this latter probability. So I don’t think you have produced a compelling counterexample yet (though I expect you are right that one exists).